Group homomorphism

A group homomorphism preserves multiplication:

ϕ (x y) = ϕ (x) ϕ (y)

.

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Surjective group homomorphism

A group homomorphism

f : G \to H

is surjective when every element of

H

is

f (g)

for some

g \in G

.

Covering homomorphism

A covering homomorphism is a surjective homomorphism of topological groups whose underlying map is a covering space. Its discrete kernel is the fiber over the identity.

Homomorphism between cyclic groups

A homomorphism from

C_{n}

is determined by the image of one generator; a surjection

C_{n} \to C_{m}

exists exactly when

m ∣ n

.

Kernel of a group homomorphism

The kernel

{g : ϕ (g) = 1}

is a normal subgroup of the domain.

Image of a group homomorphism

The image of a group homomorphism

ϕ : G \to H

is the subgroup

{ϕ (g) : g \in G}

of

H

.

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Group homomorphism by

Wikipedia Bot  1

A group homomorphism is a function between two groups that preserves the group structure.

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Group homomorphism by

Ciro Santilli  40 Updated 2025-07-16

Like isomorphism, but does not have to be one-to-one: multiple different inputs can have the same output.

The image is as for any function smaller or equal in size as the domain of course.

This brings us to the key intuition about group homomorphisms: they are a way to split out a larger group into smaller groups that retains a subset of the original structure.

As shown by the fundamental theorem on homomorphisms, each group homomorphism is fully characterized by a normal subgroup of the domain.

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